Differentiations and Diversions

Asymptotic procedures, such as generating slowness

corrections to geometric phases, involve successive differentiations. For a

large class of functions, the universal attractor of the differentiation map is,

when suitably scaled, locally trigonometric/exponential; nontrivial examples

illustrate this. For geometric phases, the series must diverge, reflecting the

exponentially small final transition amplitude. Evolution of the amplitude

towards this final velue depends sensitively on the representation used. If

this is optimal, the transition takes place rapidly and universally across a

Stokes line emanating from a degeneracy in the complex time plane. But some

Hamiltonian ODE systems do not generate transitions; this is because the

complex-plane degeneracies have a peculiar structure, for which there is no

Stokes phenomenon.  Oscillating high

derivatives (asymptotic monochromaticity) and superoscillations (extreme

polychromaticity) are in a sense opposite mathematical phenomena.